$p$-adic rigidity of eigenforms of infinite slope
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914728017657856 |
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| author | Conti, Andrea |
| author_facet | Conti, Andrea |
| contents | We give a notion of $p$-adic families of Hecke eigenforms that allows for the slope of the forms be infinite at $p$. We prove that, contrary to the case of finite slope when every eigenform lives in a Hida or Coleman family, the only families of infinite slope are either twists of Hida or Coleman families with Dirichlet characters of $p$-power conductor, or non-ordinary families with complex multiplication. Our proof goes via a local study of deformations of potentially trianguline Galois representations, relying on work of Berger and Chenevier, and a global input coming from an analogue of a result of Balasubramanyam, Ghate and Vatsal on a Greenberg-type conjecture for families of Hilbert modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16918 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $p$-adic rigidity of eigenforms of infinite slope Conti, Andrea Number Theory 11F33, 11F80 We give a notion of $p$-adic families of Hecke eigenforms that allows for the slope of the forms be infinite at $p$. We prove that, contrary to the case of finite slope when every eigenform lives in a Hida or Coleman family, the only families of infinite slope are either twists of Hida or Coleman families with Dirichlet characters of $p$-power conductor, or non-ordinary families with complex multiplication. Our proof goes via a local study of deformations of potentially trianguline Galois representations, relying on work of Berger and Chenevier, and a global input coming from an analogue of a result of Balasubramanyam, Ghate and Vatsal on a Greenberg-type conjecture for families of Hilbert modular forms. |
| title | $p$-adic rigidity of eigenforms of infinite slope |
| topic | Number Theory 11F33, 11F80 |
| url | https://arxiv.org/abs/2403.16918 |